Non-Hermitian degeneracies of internal–external mode pairs in dielectric microdisks
1. INTRODUCTION
Openness is a generic property of realistic quantum and wave systems. Hence, their dynamics is in general described by a non-Hermitian Hamiltonian with complex eigenvalues. An interesting feature of this non-Hermitian physics is the exceptional point (EP) in parameter space where both the eigenvalues (complex frequencies) and the corresponding eigenstates (modes) coalesce [13" target="_self" style="display: inline;">–
Optical microdisk cavities are ideal model systems to study non-Hermitian physics [20]. In recent years, it was demonstrated that EPs in microcavities can be caused by several mechanisms, e.g., by boundary deformations [12,21–
Along with the internal modes, there is another class of optical modes known as “external modes,” which have a large decay rate and almost no intensity inside the cavity. The internal and external modes have been classified in previous studies [2830" target="_self" style="display: inline;">–
In spite of the pioneer works proving the interactions between internal and external modes [30], there have been no investigations conceiving the possibility of EPs associated with external modes. Therefore, the aim of this paper is to reveal the existence of such EPs involving internal and external mode pairs. To demonstrate this, we introduce two kinds of systematic perturbations of a circular cavity: (i) a deformation of the cavity’s boundary, and (ii) an absorbing material.
The paper is organized as follows. In Section
2. MODES IN THE DIELECTRIC MICRODISK
For the convenience of the reader, we repeat some known facts of optical modes in quasi-two-dimensional microdisks [20]. The damped time-harmonic solutions of Maxwell’s equations in the dielectric microdisk cavity with effective refractive index
Through the separation of variables in polar coordinates, the optical modes in the circular cavity with radius
Figure
Fig. 1. (a) Real and (b) imaginary part of scaled frequency of the modes in the microdisk obtained by Eq. (1 ) for TE polarization with , as a function of azimuthal mode number . Dots (·), crosses (+), and open circles (∘) mark internal modes, external modes, and corresponding to the Brewster angle, respectively.
3. WEAK AND STRONG COUPLING OF MODES IN CIRCULAR CAVITIES
In this section, we consider a circular cavity with a real-valued refractive index
Fig. 2. Degree of degeneracy of mode pairs and given by Eq. (3 ) as a function of azimuthal mode number and refractive index in the circular cavity. The arrowed curve separates the weak and strong coupling regimes. and the color code are in log scale from (black) to (white) . The refractive index is sampled with 500 points from to .
It is an interesting fact that even in the circular cavity, signatures of an EP can be observed in the parameter space spanned by the real-valued refractive index and the radial mode number
The weak and strong coupling regimes are readily appreciable through a toy model of a non-Hermitian Hamiltonian where
In Fig.
Fig. 3. (a) Real and (b) imaginary part of in the circular cavity as a function of refractive index with a fixed azimuthal mode number undergoing strong (i, ii) and weak [(iii, iv) and (v, vi)] coupling between and with different radial mode numbers , 4, and 5. Solid-black and dashed-orange curves are and , respectively. Solid-blue arrows in (a) and (b) guide the trajectory of for increasing . Dotted-blue vertical arrows indicate the change of a radial mode number of (frequency of the nearest internal mode to the Brewster mode) from 3 to 4. The right panels show intensities of the modes marked by the same labels as in (a) and (b). The white-quarter circular and sky-blue oscillating curves superimposed on the right panels are the cavity boundaries and , respectively.
Figure
Fig. 4. (a) Real and (b) imaginary part of and as a function of refractive index with a fixed azimuthal mode number . The short segmented orange curves with a steeper slope in (a) and upper fluctuating orange curves in (b) belong to with different radial mode number . The mode number increases from left to right. The black curves with the more gentle slope in (a) and the lower fluctuating black curves in (b) belong to . Two examples and 18 of the internal mode with frequency are indicated in (a). Thin solid-green and dashed-red curves connect the values of and at which the real or the imaginary part of them crosses. The vertical-blue arrow separates the regions of strong and weak coupling at .
Up to now, the EPs in the circular cavity, with real-valued refractive index
4. DEFORMED CAVITY WITH REAL-VALUED REFRACTIVE INDEX
We consider the microflower cavity [35,36] where
Since our goal is to efficiently couple the Brewster mode with frequency
Fig. 5. Illustration of the cavity boundary in Eq. (6 ) with and . The gray shaded region bounded by the corrugated black curve depicts the deformed cavity, while the region bounded by the dashed red curve is the undeformed circle ( ). and are the refractive indices of the interior and exterior of the cavity, respectively.
We calculate
Fig. 6. (a) Real and (b) imaginary part of the frequencies of the modes in the microflower cavity as a function of deformation parameter and refractive index with fixed . Note that decreases from left to right. Labels iii and iv are the same as in Fig. 3 . Orange curves connecting the points marked by numbers from 1 to 12 show the Riemann surface topology around the EP. Blue curves are the branch-cut in Eq. (7 ).
Fig. 7. Intensity mode pattern in the microflower cavity corresponding to the marked points in Fig. 6 with the same labels. The white circular and corrugated curves are the cavity boundaries.
Fig. 8. Intensity mode pattern in the microflower cavity corresponding to the marked points in Fig. 6 with the same labels. The white corrugated circular and sky-blue oscillating curves, superimposed on the figures, are the cavity boundaries and , respectively. The red dashed horizontal line in the middle panel is the axis.
In order to ascertain EPs embedded in the Riemann surface in Fig.
The general existence of the EPs of internal and external modes is exemplified in Fig.
Fig. 9. (a) Real and (b) imaginary part of the frequency relative to of the selected internal and external modes’ frequencies as a function of deformation parameter . The EPs marked by vertical arrows are at , , and for the modes , (4,10), and (3,7), respectively.
5. CIRCULAR CAVITY WITH COMPLEX-VALUED REFRACTIVE INDEX
In the previous section, internal and external modes are tuned to cause an EP using a deformation of the cavity’s boundary. A crucial effect of such a deformation is the
In the following, we explain the scheme to derive an EP using a complex refractive index in detail. We consider as an example the internal mode
Fig. 10. (a),(c) and (b),(d) show the real and the imaginary parts of in the complex plane, respectively. The (blue and gray) dashed curves belong to of the external mode and the (orange and gray) solid curves correspond to the internal mode. In (a) and (b), the cross (empty circle) marks the initial frequency of the internal (external) mode. The EP is marked by a black dot. In (c) and (d), the vicinity of the EP is shown via . The outer thick red curve corresponds to a two-fold encircling of the EP. Thin gray dots represent the Riemann sheets of .
In Fig.
Fig. 11. (a) Real (left/red axis/dashed curves) and imaginary (right/blue axis/solid curves) parts of the wave number relative to along the parameter curve in the complex plane. The parameter curve is shown in (b), where the EP is marked as a thick black dot. parameterizes this curve starting at for real . is marked as a black cross. The corresponding mode patterns at ( ), (EP), and ( ) are shown. The color map of the intensities ranging from black to red is truncated outside the cavity at twice the maximum value inside the cavity.
The general existence of such EPs for internal–external mode pairs, through the complex refractive index, is confirmed by exemplifying several mode pairs with
Fig. 12. In (a), is shown for varying real refractive index for internal (black thick curves) and external (magenta thin curves) modes with in a circular cavity. In (b)–(e), the paths in the complex plane are shown for which two modes (one internal and one external) with have the same . The end point of each curve marks an EP. Colored dots in (a) mark the intersections as starting points for the parameter curves in (b)–(e).
In obtaining EPs through the complex-valued refractive index, some remarks are in order. First, the tracing procedure starting from the crossings in the imaginary part of
At first glance, the experimental feasibility to adjust both the real and imaginary parts of the effective refractive index over wide ranges as, e.g., in Figs.
6. SUMMARY
We revealed numerically the existence of exceptional points resulting from the coupling of two different kinds of modes, i.e., an internal (Feshbach resonance) and external mode (shape resonance), in an optical microdisk cavity. The external modes have exponentially small contributions inside the cavity and a low
7 Acknowledgment
Acknowledgment. C.-H. Y. was financially supported by the DFG (project WI1986/7-1). M. H. and J. K. acknowledge support from DFG under the Emmy Noether Programme. The authors thank J. Kreismann for fruitful discussions.
Article Outline
Chang-Hwan Yi, Julius Kullig, Martina Hentschel, Jan Wiersig. Non-Hermitian degeneracies of internal–external mode pairs in dielectric microdisks[J]. Photonics Research, 2019, 7(4): 04000464.